H-infinity control with integrator compensation for anode pressure control in a fuel cell stack

ABSTRACT

Pressure control in a fuel cell is achieved by using an H-infinity controller coupled in a feedback loop between a reactant feed gas valve and a pressure sensor on gas flows to the membrane electrode assembly of the fuel cell. To maintain pressure balance across the membrane, the pressure of the oxidant reactant is used to regulate fuel reactant flow. An integrator windup compensator manages integral windup in the H-infinity control scheme. Control weight, sensor noise weight, and performance weight matrices are incorporated into the H-infinity control model. Respective to PID control, the H-infinity model provides superior performance in the presence of high frequency feedback noise enabling use of low cost control components in the fuel cell and a minimum of EMI shielding.

FIELD OF THE INVENTION

The present invention relates to fuel cell power systems and methods forcontrolling pressure in a reactant feed gas stream to a fuel cell stackof the fuel cell power system.

BACKGROUND OF THE INVENTION

Fuel cell power systems convert a fuel and an oxidant to electricity.One fuel cell power system type of keen interest employs use of a protonexchange membrane (hereinafter “PEM”) to catalytically facilitatereaction of fuels (such as hydrogen) and oxidants (such as air/oxygen)into electricity. The PEM is a solid polymer electrolyte thatfacilitates transfer of protons from the anode to the cathode in eachindividual fuel cell of the stack of fuel cells normally deployed in afuel cell power system.

In a typical fuel cell assembly (stack) within a fuel cell power system,individual fuel cells have flow fields with inlets to fluid manifolds;these collectively provide channels for the various reactant gases andcooling fluids in the stack to flow into each cell. Gas diffusionassemblies then provide a final fluid distribution to further dispersereactant fluids from the flow field space to the reactive anode andcathode; these diffusion sections are frequently advantageously embeddedas a part of the design of collector electrodes pressing against thereactive anode and cathode.

Effective operation of a PEM requires maintenance of a small pressuredrop between the cathode (air) and anode (hydrogen) gases across thePEM; in this regard, accurate pressure control is vital to fuel cellstack performance and durability.

Control of fuel cell power systems must also resolve high frequencynoise derived from EMI (electromagnetic interference); sources of EMIare both internal from the components of the fuel cell as well asexternal, especially when the fuel cell powers a vehicle which movesfrom place to place and thereby experiences different EMI environments.

There is an ongoing desire to minimize cost in fuel cell systems. Lowcost components (such as pressure and feed control valves), however,frequently demonstrate susceptibility to EMI and also provide marginalacceptability in maintaining acceptably balanced pressures in fuel cellstacks when used with traditional PID (proportional-integral-derivative)control schemes. Components (such as pressure and feed control valves)which demonstrate good resistance to EMI and also provide acceptabilityin maintaining balanced pressures in fuel cell stacks when used withtraditional PID (proportional-integral-derivative) control schemes arenot favored for deployment because of higher cost.

What is needed is an approach to fuel cell pressure control whichprovides acceptable precision in balancing pressures across a PEM at lowcost. The present invention is directed to fulfilling this need.

SUMMARY OF THE INVENTION

The present invention provides pressure control in a fuel cell having atleast one membrane electrode assembly in reactive interface (a) to aplurality of oxidant reactant flow channels carrying an oxidant reactantand (b) to a plurality of fuel reactant flow channels carrying a fuelreactant, using: a valve disposed to control at least one reactant flowto the membrane electrode assembly; a pressure sensor disposed tomeasure pressure within the fuel cell; and an H-infinity controllercoupled in a feedback loop between the valve and the pressure sensor.

As a method, the invention operates a fuel cell having at least onemembrane electrode assembly in reactive interface (a) to a plurality ofoxidant reactant flow channels carrying an oxidant reactant and (b) to aplurality of fuel reactant flow channels carrying a fuel reactant bymeasuring pressure within the fuel cell; deriving a setpoint for atleast one reactant flow from an H-infinity control model in response topressure data from the measuring step; and regulating each reactant flowfor which the deriving step derives a setpoint.

The invention further provides that the pressure of the oxidant reactantis used to regulate fuel reactant flow.

The invention also provides for use of an integrator windup compensatorin data communication with the H-infinity controller and also for use ofa real-time computer to execute the H-infinity controller and/or thewindup compensator.

The invention further provides for incorporation of (a) a control weightmatrix, (b) a sensor noise weight matrix, and/or (c) a performanceweight matrix in the H-infinity control model.

When compared to a standard PID controller, the invention providesenhanced performance in the presence of high frequency feedback noise toprovide an improved operation of the control valve, less part-to-partactuator variation, and reduced system retuning. The invention furtherenables use of low cost control components in the fuel cell andminimizes the amount of EMI shielding needed for effective powergeneration.

Further areas of applicability of the present invention will becomeapparent from the detailed description provided hereinafter. It shouldbe understood that the detailed description and specific examples, whileindicating the preferred embodiment of the invention, are intended forpurposes of illustration only and are not intended to limit the scope ofthe invention.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention will become more fully understood from thedetailed description and the accompanying drawings, wherein:

FIG. 1A presents a first embodiment of a fuel cell power systemincorporating the present invention;

FIG. 1B presents a second embodiment of a fuel cell power systemincorporating the present invention;

FIG. 2 shows membrane electrode assembly detail in a fuel cell stackportion;

FIG. 3 shows the anode pressure due to a step response of a proportionalflow control valve;

FIG. 4 shows variance between a model of the step response according toFIG. 3 and measured data;

FIG. 5 shows a feedback control system model in a form consistent withrobust analysis;

FIG. 6 depicts a patterned schema showing management of uncertainties ina control loop model;

FIG. 7 provides a frequency response profile plot of a transfer functionrelated to a control weight matrix;

FIG. 8 provides a frequency response profile plot of a transfer functionrelated to a sensor noise weight matrix;

FIG. 9 provides a frequency response profile plot of a transfer functionrelated to a performance weight matrix;

FIG. 10 shows an exemplary feedback loop in overall linear fractionaltransformation form;

FIG. 11 provides detail in a state-space matrix of the linear fractionaltransformation form of the exemplary control problem reviewed in thisspecification; and

FIG. 12 shows detail in an H-infinity controller with an integratorwindup compensation block.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

The following description of the preferred embodiment is merelyexemplary in nature and is in no way intended to limit the invention,its application, or uses.

In utilizing a low cost proportional flow control valve to maintain abalanced pressure across a PEM, the anode pressure setpoint and/or fuelflow to the anode is determined from the cathode pressure against a userselected pressure set-point. In the past, standard PID control has beenused for such control along with supporting feedforward loops andsoftware filters. However, H-infinity (H^(∞)) control of thisapplication is shown herein to be well suited as a robust controlapproach. Details in the methodology of H-infinity control areestablished in the art and are appreciated from a study of “Essentialsof Robust Control” by Kermin Zhou and John C. Doyle (Prentice Hall,1998). The discussion of the preferred embodiments also references twoadditional concepts as further detailed in the following two paragraphs.

The discussion of the preferred embodiments references the concept of“steady-state” operation. As used herein, “steady-state” operation or“steady state” is considered a situation where (1) a process isdynamically regular and uniform in its operation over a time interval,(2) momentum, mass, and energy entities flowing into the process areessentially equal to the momentum, mass, and energy entities flowing outof the process (excepting anticipated accumulations), and (3)accumulations of momentum, mass, and energy within the process areessentially not occurring unless they are explicitly expected andfactored into the relevant dynamic model. Mathematical solutions of thebalances with respect to the status of steady state operation need toalso accommodate expected chemical reactions. Steady state operation ofa system is an issue of importance to the present invention sincecertain of the modeling equations are based upon the presumption thatreal-time input data used in a specific instance of a control decisionhave a collective associated steady state relationship. A system in“steady state” therefore, has attributes of dynamic balance, stability,steadiness, and equilibrium.

The concept of real-time computer process control is also a useful termin understanding the preferred embodiment. As used herein, real-timecomputer processing is broadly considered as a method of computerprocessing in which an event causes a given reaction within an actualtime limit and wherein computer actions are specifically controlledwithin the context of and by external conditions and actual times. As anassociated clarification in the realm of process control, real-timecomputer controlled processing relates to the performance of associatedprocess control logical, decision, and quantitative operations intrinsicto a process control decision program functioning as part of acontrolled apparatus implementing a process (such as the fuel cellbenefiting from the present invention) wherein the process controldecision program is periodically executed with relatively highfrequency—e.g., having a period of between 20 ms and 2 sec for highlytactical control, or on the order of 10 to 100,000 times the period ofthe associated tactical control decision frequency for “on line”real-time advanced control routines, simulators, and optimizers, withoutlimitation. The larger period for advanced control routines, simulators,and optimizers is frequently necessary to accommodate the substantialcomputer calculations which must be performed within one decision cycleof the advanced control routine, simulator, or optimizer. With furtherregard to the time period during which the process control decisionprogram is periodically executed, some operations are optionallyperformed on a multiple of the process control decision programexecution period needed for computation time; this less frequentoperation period is usually adopted for purposes related to tuning,sensitivity, and efficient resource utilization.

The invention is further understood with reference to a generic fuelcell power system. Therefore, before further describing the invention indetail, a general overview of the types of power systems within whichthe present invention operate is provided. Reference is made tohydrogen-containing as having relatively high hydrogen content. Theinvention is hereafter described in the context of a fuel cell fueled byan H₂-containing reformate regardless of the method by which suchreformate is made. It is to be understood that the principles embodiedherein are applicable to fuel cells fueled by H₂ obtained from anysource, including reformable hydrocarbon and hydrogen-containing fuelssuch as methanol, ethanol, gasoline, alkaline, or other aliphatic oraromatic hydrocarbons.

A first preferred system 100 a illustrated in FIG. 1A includes hydrogensource 102 such as a hydrogen storage tank and oxidant source 104 suchas ambient air provided via a pump or compressor (not shown). Hydrogensource 102 directs H₂-containing feed stream 120 to the anode side offuel cell 122. Oxidant source 104 directs O₂-containing feed stream 124to the cathode side of fuel cell 122. Anode exhaust (or effluent) 126 isdischarged from the anode side of fuel cell stack system 122. Cathodeexhaust (or effluent) 128 is discharged from the cathode side of fuelcell stack system 122. Pressure of cathode exhaust 128 from the cathodeside of fuel cell stack system 122 is measured by pressure sensor 160.

A second preferred system 100 b illustrated in FIG. 1B includes a fuelprocessor 112 for catalytically reacting a reformable hydrocarbon fuelstream 114, and water in the form of steam from a water stream 116. Insome fuel processors, air is also used in a combination partialoxidation/steam reforming reaction. In this case, fuel processor 112also receives an air stream 118. The fuel processor 112 contains one ormore reactors wherein the reformable hydrocarbon fuel in stream 114undergoes dissociation in the presence of steam in stream 116 and air instream 118 to produce the hydrogen-containing reformate exhausted fromfuel processor 112 in reformate stream 120. Fuel processor 112 typicallyalso includes one or more downstream reactors, such as water-gas shift(WGS) and/or preferential oxidizer (PrOx) reactors that are used toreduce the level of carbon monoxide in reformate stream 120 toacceptable levels, for example, below 20 ppm.

Anode exhaust (or effluent) 126 is discharged from the anode side offuel cell stack system 122 and may contain some unreacted hydrogen.Cathode exhaust (or effluent) 128 is discharged from the cathode side offuel cell stack system 122 and may contain some unreacted oxygen.Pressure of cathode exhaust 128 from the cathode side of fuel cell stacksystem 122 is measured by pressure sensor 160. These unreacted gasesrepresent additional energy recovered in combustor 130, in the form ofthermal energy, for various heat requirements within power system 100.Specifically, a hydrocarbon fuel 132 and/or anode effluent 126 arecombusted, catalytically or thermally, in combustor 130 with oxygenprovided to combustor 130 either from air in stream 134 or from cathodeeffluent stream 128, depending on power system 100 operating conditions.Combustor 130 discharges exhaust stream 154 to the environment, and theheat generated thereby is directed to fuel processor 112 as needed.

In both embodiments illustrated in FIGS. 1A and 1B, H₂-containingreformate 120 is fed through control valve 162 into the anode chamber offuel cell stack system 122. Control valve 162 may be either an analogcontrol valve or a solenoid spring return valve similar to a fuelinjector valve with a 100 Hz duty cycle frequency. Concurrent with thefeeding of H₂-containing reformate 120 through control valve 162 intothe anode chamber of fuel cell stack system 122, oxygen in the form ofair in stream 124 is fed into the cathode chamber of fuel cell stacksystem 122. The hydrogen from reformate stream 120 and the oxygen fromoxidant stream 124 react in fuel cell stack system 122 to produceelectricity.

Real-time computer 164 effects control of valve 162 in response to asignal from at least pressure sensor 160. That is to say the anode feedgas is controlled through use of real-time computer 164 with respect tothe pressure of the cathode oxidant gas in fuel cell 122, although otherparameters may also be utilized in the control of the anode feed gas.Controller logic 166 is provided for execution in real-time by computer164. As presently preferred, controller logic 166 is also denoted as“software” and/or a “program” and/or an “executable program” withinreal-time computer 164 as a data schema holding data and/or formulaeinformation and/or program execution instructions. Controller logic 166is, in a preferred embodiment, machine code resident in the physicalmemory storage (i.e., without limitation, random access memory having“RAM” as an indicator, read only memory having “ROM” as an indicator, ora computer disk) of computer 164. Controller logic 166 is preferablyderived from a source language program compiled to generate the machinecode. The physical memory storage is in electronic data communicationwith a central processing unit (CPU) of computer 164 which reads datafrom the physical memory, computationally modifies read data intoresultant data, and writes the resultant data to the physical memory.Computer 164 also read signals from sensor 160 and effects signals tovalve 162 according to the provisions of controller logic 166.

The fuel cell power systems described abve include a hydrogenstorage-based system or a fuel reforming system. Thus, a skilledpractitioner will recognize that the present invention has applicationto a variety of system which obtain fuel from diverse sources. In thisregard, the manner in which the fuel is generated does not impact thepresent invention or its application into a fuel cell power system.

Turning now to FIG. 2, a partial PEM fuel cell stack 200 of fuel cellstack system 122 is schematically depicted as having a pair of membraneelectrode assemblies (MEAs) 208 and 210 separated from each other by anon-porous, electrically-conductive bipolar plate 212. Each of MEAs 208,210 have a cathode face 208 c, 210 c and an anode face 208 a, 210 a.MEAs 208, 210 and bipolar plate 212 are stacked together betweennon-porous, electrically-conductive, liquid-cooled end plates 214 and216. Plates 212, 214, 216 each include respective flow fields 218, 220,222 established from a plurality of flow channels formed in the faces ofthe plates for distributing fuel and oxidant gases (i.e., H₂ & O₂) tothe reactive faces of MEAs 208, 210. Nonconductive gaskets or seals 226,228, 230, 232 provide sealing and electrical insulation between theseveral plates of fuel cell stack 200.

Porous, gas permeable, electrically conductive sheets 234, 236, 238, 240press up against the electrode faces of MEAs 208, 210 and serve asprimary current collectors for the respective electrodes. Primarycurrent collectors 234, 236, 238, 240 also provide mechanical supportsfor MEAs 208, 210, especially at locations where the MEAs are otherwiseunsupported in the flow field. Bipolar plate 214 presses up againstprimary current collector 234 on cathode face 208 c of MEA 208, bipolarplate 216 presses up against primary current collector 240 on anode face210 a of MEA 210, and bipolar plate 212 presses up against primarycurrent collector 236 on anode face 208 a of MEA 208 and against primarycurrent collector 238 on cathode face 210 c of MEA 210.

An oxidant gas such as air/oxygen is supplied to the cathode side offuel cell stack 200 from air source 118 and line 124 via appropriatesupply plumbing 242. In a preferred embodiment, air is supplied to thecathode side from the ambient. A fuel such as hydrogen is supplied tothe anode side of fuel cell 200 from fuel source 270 via appropriatesupply plumbing 244. In a preferred embodiment, the fuel source issupplied from a reformer via line 120 after catalytically dissociatinghydrogen from hydrocarbon fuel 114.

Exhaust plumbing (not shown) for both the H₂ and O₂/air sides of MEAs208, 210 is also provided for removing anode effluent from the anodeflow field and the cathode effluent from the cathode flow field. Coolantplumbing 250, 252 is provided for supplying and exhausting liquidcoolant to bipolar plates 212, 214, 216, as needed.

It is to be noted that fuel cell stack 200 shows two fuel cells withplate 212 being shared between the two fuel cells and plates 214, 216being shared between one of the shown fuel cells and, in each case,another fuel cell not depicted in FIG. 2.

Turning now to further detail in controller logic 166 of real-timecomputer 164 and with reference to FIG. 1, fuel cell power system 100uses proportional flow control valve 162 to control reactant feed gasflow, and pressure sensor 160 is used as a feedback sensor measuringcathode gas pressure of fuel cell 122. A robust H-infinity controller iseffected in controller logic 166 in a feedback loop between valve 162and pressure sensor 160.

In overview, the first step of the present invention is to obtain afundamental dynamic response model from the control loop defined frompressure sensor 160, real-time computer 164, and control valve 162. Aswill be described further herein, a first-order model empiricallycharacterizes the dynamic relationship of the feedback loop betweenvalve 162 and pressure sensor 160 in fuel cell power system 100 for theexemplary control loop and derived H-infinity robust controller. In thisregard, a standard discrete time system identification technique,AutoRegressive eXogeneous or ARX, determines the first-orderrelationship from numerous open-loop responses of pressure 160 to stepchanges in control valve 162 and derives a set of models encompassingthe essential full range of anticipated behavior for fuel cell powersystem 100. Response model uncertainties may be further determined basedon the standard deviation of the models found during each responsemeasurement.

The second step relates to development of the H-infinity data schema forthe H-infinity controller. The dynamic response model with uncertainties(from the first step of the overview) is combined with various weightingdata. For example, dynamic response noise “weighting” data (e.g., anoise weight matrix) derived from measurements of known high frequencyEMI (electromagnetic interference) feedback noise is combined with thecombined data derived from the design and measurements of the closedcontrol loop and configured into a Linear Fractional Transformation(LFT) framework. Dynamic controller response “weighting” data (e.g. acontrol weight matrix) and dynamic response plant performance“weighting” data (e.g. a performance weight matrix) may be furtherincorporated into the LFT framework. An H-infinity data schema for anH-infinity real-time controller is then calculated from the LFTframework. Thus, the result of the first step is a dynamic responsemodel which includes data set(s) derived from the design andmeasurements of the closed control loop.

Insofar as a controller derived solely from the H-infinity data schemawill have an integral windup effect, a integrator compensation gain mayalso be effected in the H-infinity controller of controller logic 166 inreal-time computer 164 to provide an integrator windup compensationblock in controller logic 166, and thereby provide real-timemodification of the output from real-time calculations of the H-infinitydata schema to compensate for integrator windup in the control loop andmaintain control valve 162 in an immediately responsive status.

Lastly, the robust control data schema and integrator windup compensatorare operably incorporated in controller logic 166 of real-time computer164 which is employed to control fuel cell power system 100.

Turning now to greater detail in implementing the steps described in theabove overview, FIGS. 3 and 4 show step response 300 and empirical dataand model comparison 306, respectively the curves illustrated in graphs300, 400 may be derived (either directly of a given fuel cell or via aproximate pressure vessel physical model thereof) from the exemplaryfeedback loop between valve 162 and pressure sensor 160 in fuel cellpower system 100 through use of a conventional system identificationtechnique for discrete data such as AutoRegressive output with exogenousinput, or ARX. Numerous textbooks have been written on this subject, agood discussion of detail is presented in “Applied SystemIdentification” by Jer-Nan Juang (PTR Prentice Hall, 1994). Aspreviously noted, details in the methodology of H-infinity control areestablished in the art and are appreciated from a study of “Essentialsof Robust Control” by Kermin Zhou and John C. Doyle (Prentice Hall,1998).

FIG. 3 shows step response 302 from step change 304 indicating positionof control valve 162. This data indicates a first-order relationship forthe exemplary feedback loop. A discrete time transfer function istherefore derived as:

$\begin{matrix}{\frac{y(z)}{u(z)} = \frac{\beta}{z + \alpha}} & \left( {{Eq}.\mspace{14mu} 1} \right)\end{matrix}$Conversion of Equation 1 to sampled time yields:y(k)=−αy(k−1)+βu(k−1)  (Eq. 2)Equation 2 is reformatted into matrix form with α and β in Equation 2derived from a least-squares solution technique. The further derivedz-domain ARX model is:

$\begin{matrix}{\frac{\hat{P}}{ValveCmd} = \frac{- 0.0371}{z - 0.9775}} & \left( {{Eq}.\mspace{14mu} 3} \right)\end{matrix}$

FIG. 4 shows a first-order fit 306 to normalized pressure response datafor the exemplary feedback loop respective to Equation 3. Note that a100% position for control valve 162 is a completely closed configurationfor the valve. Since pressure sensor 160 has a non-zero IC, a bias ofapproximately-260 kPa is used to transition the ordinate of FIG. 3 tothe ordinate of FIG. 4. First-order model prediction 308 as plottedagainst experimental data 310 is sufficiently accurate with a data minusmodel variance of σ² _(data-model)=2.8084.

In order to proceed with continuous-time robust analysis and H-infinitycontrol data schema development, the z-domain transfer function ofEquation 3 is converted to s-domain form as:

$\begin{matrix}{\frac{\hat{P}}{ValveCmd} = \frac{- 0.3749}{s + 0.2273}} & \left( {{Eq}.\mspace{14mu} 4} \right)\end{matrix}$The conversion to Equation 4 assumes a 1^(st) order transfer function, asample rate of 10 Hz, and a zero-order hold.

The ARX modeling procedure is repeated for several step changes in valveposition with Table 1 showing the model coefficients α and β derivedfrom each step change.

TABLE 1 System Identification. Start Step Start Time End Step End TimeDenominator Numerator Error Model Order Input Output Valve sec Valve secalpha beta Variance 1st Valve P 0 13.4 100 143.5 −0.982 −0.0319 45.38551st Valve P 100 143.5 10 188.1 −0.978 −0.0371 2.8084 1st Valve P 10188.1 0 281.3 −0.985 −0.0260 39.4215 1st Valve P 0 281.3 100 284.8−0.952 −0.0360 0.0319 1st Valve P 100 284.8 20 344.9 −0.977 −0.03791.0573 1st Valve P 20 344.9 100 441.2 −0.979 −0.0363 2.5137 1st Valve P100 441.2 30 500.6 −0.976 −0.0415 0.7038 1st Valve P 30 500.6 100 588.3−0.978 −0.0340 2.1465 1st Valve P 100 588.3 40 676.7 −0.973 −0.04120.3709 1st Valve P 40 676.7 100 741.3 −0.975 −0.0317 1.6924 1st Valve P100 741.3 50 814.4 −0.972 −0.0365 0.3134 1st Valve P 50 814.4 100 871.6−0.971 −0.0264 0.8339 1st Valve P 100 871.6 60 923.9 −0.965 −0.03260.0914 1st Valve P 60 923.9 100 976.6 −0.957 −0.0203 0.3657 1st Valve P100 976.6 70 1057.0 −0.944 −0.0247 0.0638 1st Valve P 70 1057.0 1001100.6 −0.959 −0.0130 0.1473 1st Valve P 100 1100.6 80 1130.8 −0.935−0.0193 0.0155 1st Valve P 80 1130.8 0 1152.3 −0.841 −0.0149 0.0159 1stValve P 0 1152.3 80 1220.5 −0.982 −0.0382 7.1789 1st Valve P 80 1220.520 1269.3 −0.977 −0.0482 0.6847 1st Valve P 20 1269.3 80 1325.5 −0.979−0.0469 5.0218 1st Valve P 80 1325.5 40 1373.3 −0.975 −0.0552 0.2094 1stValve P 40 1373.3 80 1429.4 −0.975 −0.0470 2.0239 1st Valve P 80 1429.460 1472.4 −0.970 −0.0568 0.0467 1st Valve P 60 1472.4 80 1538.0 −0.966−0.0357 0.2319 1st Valve P 80 1538.0 100 1604.1 −0.957 −0.0449 0.0814FULL DATA Nominal −0.9646 −0.0352 4.3637 StandardDev 0.0274 0.011111.1366

As should be apparent, the model results are sufficiently accurate overnumerous open-loop responses of pressure measurements from sensor 160 tostep changes in control valve 162 for deriving a set of models whichencompass the essential full range of anticipated behavior for fuel cellpower system 100 with (a) further confirmation of the first-orderresponse nature of the system over the full range of anticipatedbehavior and (b) a model minus data variance of less than 10 for nearlyall cases. Once a set of models is found, a nominal model is derivedaccounting for modeling uncertainties with uncertainty parameters andone standard deviation. The nominal discrete model with theuncertainties (1σ) from Equation 4 and Table 1 is:

$\begin{matrix}{\frac{\hat{P}}{ValveCmd} = \frac{- \left( {0.0352 \pm 0.0111} \right)}{z - \left( {0.9646 \pm 0.0274} \right)}} & \left( {{Eq}.\mspace{14mu} 5} \right)\end{matrix}$Converting the discrete domain model of Equation 5 to a derivedcontinuous model yields:

$\begin{matrix}{\frac{\hat{P}}{ValveCmd} = {\frac{\beta}{s + \alpha} = \frac{- \left( {0.3569 \pm 0.1084} \right)}{s + \left( {0.3645 \pm 0.2882} \right.}}} & \left( {{Eq}.\mspace{14mu} 6} \right)\end{matrix}$

Once the dynamic response model of Equation 6 is defined, the system isformulated into a form consistent with robust analysis. This processentails (a) “pulling-out” the uncertainties in the control loop, (b)defining weight matrices, and (c) formulating the derived models anddata into a Linear Fractional Transformation (LFT).

FIG. 5 shows feedback control system model 500 for the pressure controlsystem in fuel cell 122 in a form consistent with robust analysis. Modelblock 502, control weight matrix 504 (W_(u)), sensor noise weight matrix506 (W_(n)), performance weight matrix 508 (W_(e)), system disturbanceboundary 512 (W_(do)), system disturbance boundary 510 (W_(di)), andController 166 (as converted to physical flow via control valve 162) allinterrelate as shown to feed fuel (hydrogen) and affect cathode pressurecontrol within fuel cell 122.

Turning now to FIG. 6, the uncertainties in the control loop are “pulledout” according to patterned schema 600. With reference to Equation 6,FIG. 6 illustrates how the uncertainties in the dynamic response modelare removed from the nominal feedback loop. The considerationssummarized by FIG. 6 allow the amount of uncertainty in the feedbackloop to be determined by δ_(α) at model block 602 and δ_(β) at modelblock 604. These parameters are allowed to take a form between ±1. Aconsideration of FIG. 6 indicates that if δ_(α) and δ_(β) are set tozero, a nominal feedback loop results. The offsets are derived fromnormalizing the initial conditions to zero for the ARX model.

After uncertainties in the control loop have been “pulled out”, weightmatrices to provide “weighting” in the real-time computations ofcontroller logic 166 are defined. Three weight matrices are used in theexemplary feedback loop to address three different issues—controlleroutput, noise input, and plant performance output. These matrices areidentified in FIG. 5 as control weight matrix 504 (W_(u)), sensor noiseweight matrix 506 (W_(n)), and performance weight matrix 508 (W_(e)).

Control weight matrix 504 (W_(u)) is derived from:

$\begin{matrix}{W_{u} = \frac{s + 0.1}{{0.1s} + 10}} & \left( {{Eq}.\mspace{14mu} 7} \right)\end{matrix}$The control weight matrix, W_(u), is defined to provide full performanceat low frequencies (<0.1 rad/sec) but reduced performance at higherfrequencies (>100 rad/sec). FIG. 7 provides a frequency response profileplot 700 of the transfer function in Equation 7.

Sensor noise weight matrix 506 (W_(n)) is derived from:

$\begin{matrix}{W_{n} = \frac{10\left( {s + 10} \right)}{s + 1000}} & \left( {{Eq}.\mspace{14mu} 8} \right)\end{matrix}$Sensor noise weight matrix (W_(n)) is defined to provide an essentiallylow amount of corruption from low frequency (<10 rad/sec) noise with atransition to increased corruption at higher frequencies (>1000rad/sec). In this regard, pressure sensor 160 is highly susceptible tohigh frequency EMI noise. FIG. 8 provides a frequency response profileplot 800 of the transfer function in Equation 8. Note that, atfrequencies greater than 1000 rad/sec, the signal from pressure sensor160 is highly corrupted by as much as +/−10 kPa.

Performance noise weight matrix 508 (W_(e)) is derived from:

$\begin{matrix}{W_{e} = \frac{s + 0.8}{{0.08s} + 0.0008}} & \left( {{Eq}.\mspace{14mu} 9} \right)\end{matrix}$Performance weight matrix 508 (W_(e)) is defined to provide anessentially ideal dynamic response model at near steady-state (<0.1rad/sec) with a transition to a more substantial deviation from theideal response model at higher frequencies. As will be apparent from thebasis of the modeling, the ideal plant is chosen as a simple first ordersystem per model block 502. FIG. 9 provides a frequency response profileplot 900 of the transfer function in Equation 9.

After “pulling-out” of the uncertainties in the control loop anddefinition of weight matrices has been achieved, the derived models anddata are formulated into a Linear Fractional Transformation (LFT). Theexemplary feedback loop is shown in overall LFT form 1000 in FIG. 10.Difference block 1002, model block 1004, and controller block 1006 allinterrelate as shown in LFT form 1000.

In the context of FIG. 5 and through use of control weight matrix 504(W_(u)), sensor noise weight matrix 506 (W_(n)), and performance weightmatrix 508 (W_(e)), the following series of equations and therelationship illustrated in FIG. 11 formalize the exemplary closed-loopcontrol system of this specification into LFT form. FIG. 11 shows LFTmatrix 1100 having quadrants A, B, C, and D respectively designatingdomains relative to companion variables with the same A, B, C, and Dprimary symbols in Equations 11, 12, 13, 14, 20, and 21.e=r−y _(p) −y _(n)  (Eq. 10){dot over (x)} _(n) =A _(n) x _(n) +B _(n) n  (Eq. 11)y _(n) =C _(n) x _(n) +D _(n) n  (Eq. 12){dot over (x)} _(u) =A _(u) x _(u) +B _(u) u  (Eq. 13)y _(u) =C _(u) x _(u) +D _(u) u  (Eq. 14){dot over (x)}_(p)=u_(p)  (Eq. 15)u _(p) =u−100−{overscore (α)}x _(p) −u _(α)  (Eq. 16)y _(p) ={overscore (β)}x _(p) +u _(β)+100  (Eq. 17)y _(α) ={circumflex over (α)}x _(p)  (Eq. 18)y _(β) ={circumflex over (β)}x _(p)  (Eq. 19){dot over (x)} _(e) =A _(e) x _(e) +B _(e) e  (Eq. 20)y _(e) =C _(e) x _(e) +D _(e) e  (Eq. 21)[y _(α) y _(β) y _(u) y _(e) e] ^(T) =G[u _(α) u _(β)1r n u]  (Eq. 22)

As should be apparent from the steps leading to the above-described LFTformalization, concerns related to uncertainty in operation of fuel cell122, sensor 160 noise, and responsiveness at different states ofoperation are effectively incorporated into the LFT formalization.

Once the system is in LFT form, an H-infinity control data schema isderived. In this regard, the MATLAB μ-analysis toolbox add-on availablefrom The Mathworks, Inc. of Natick, Mass. is convenient for solving theLFT and Algebraic Riccati Equation to calculate a robust controller. Forthe exemplary control loop, the resulting control statement for theH-infinity control data schema is

$\begin{matrix}{K = \frac{{1.6814s^{3}} - {1854s^{2}} - {169550s} - 87308}{s^{4} + {1007s^{3}} + {12578s^{2}} + {77073s} + 2.1017}} & \left( {{Eq}.\mspace{14mu} 23} \right)\end{matrix}$

In order to analyze robust stability and nominal performance in asimulator, controller K is incorporated into G to yieldG_(p)=GK  (Eq. 24)Wherein:

$\begin{matrix}{\left\lbrack \begin{matrix}y_{\alpha} \\y_{\beta} \\y_{u} \\y_{e}\end{matrix} \right\rbrack = {{G_{p}\left\lbrack \begin{matrix}u_{\alpha} \\u_{\beta} \\1 \\r \\n\end{matrix} \right\rbrack} = {\left\lbrack \begin{matrix}G_{p11} & G_{p12} \\G_{p21} & P_{p22}\end{matrix} \right\rbrack\left\lbrack \begin{matrix}u_{\alpha} \\u_{\beta} \\1 \\r \\n\end{matrix} \right\rbrack}}} & \left( {{Eq}.\mspace{14mu} 25} \right)\end{matrix}$As will further be appreciated by those of skill, robust stability,nominal performance, and μ-analysis are recommended for application tothe designed robust controller. Simulated results of robust controlleruse should also be considered for analysis to observe the performance ofthe control data schema.

Turning now to FIG. 12 (showing further detail in the H-infinitycontroller), integrator windup compensation block 1202 in controllerlogic 166 (illustrated in FIG. 1) is included to “stall” output fromH-infinity data schema 1006 (including controller K of Equation 11) incases where controller saturation occurs during real-time execution ofschema 1006 by real-time computer 164. In this regard, a controllerderived solely from the H-infinity data schema may have an integralwindup effect. Integrator windup compensation block 1202 effectsreal-time modification of the output from real-time calculations ofH-infinity data schema 1006 to compensate for integrator windup in thecontrol loop and maintain control valve 162 in an immediately responsivestatus. In this regard, controller saturation influence is input viablock 1206 into memory section 1204 in computer 164. Memory section 1204inputs a value to integrator windup compensation block 1202 so thatintegrator windup compensation block 1202 outputs a modifying value forH-infinity data schema 1006.

A number of benefits are derived from the use of H-infinity controllerin a fuel cell power system in accordance with the present invention.These are appreciated in general comparison to a PID(proportional-integral-derivative) controller. The robust controlapproach of the described H-infinity controller provides superiorperformance in the presence of high frequency feedback noise (e.g.,without limitation, +/−10 kPa high frequency EMI) when compared to astandard PID controller. Since a PID control signal will vary moredramatically than the control signal from an H-infinity controller, asignificantly improved operation of control valve 162 accrues from theuse of the H-infinity controller. A PID control strategy also requiresretuning of its affiliated gains in the event of a change in systemdynamics; this is not needed with the H-infinity controller. Morever,part-to-part actuator variation is also less in the H-infinitycontroller case.

In the described H-infinity controller, fuel cell uncertainties aredirectly incorporated into the problem formulation. In this regard, aprimary problem in control development is that modeled systems thatchange over time frequently render an original dynamic response modelunacceptable for control over the long term which is diminished by useof H-infinity control.

Signal noise is also incorporated into the formulation of the describedH-infinity controller. In this regard, signal corruption is typicallyquantified according to its frequency response. Filtering of thesefrequencies is applied to the control data schema in the H-infinitycontroller. This enables deployment of low cost sensors and valves witha saving to overall system cost. Because the H-infinity controller has ahigh tolerance to EMI, wiring and packaging needs are also minimizedrespective to EMI shielding.

The description of the invention is merely exemplary in nature and,thus, variations that do not depart from the gist of the invention areintended to be within the scope of the invention. For example, thepreferred embodiment has been described in reference to measuring thepressure of the cathode exhaust stream and controlling the anode feedstream. However, one skilled in the art will appreciate that thelocation of the pressure measurement may be varied. Likewise, the sidesof the fuel cell for measurement (cathode side) and control (anode side)may be interchanged. Such variations are not to be regarded as adeparture from the spirit and scope of the invention.

1. A fuel cell system comprising: a fuel cell with at least one membraneelectrode assembly in reactive interface with an oxidant reactant on oneface thereof and a fuel reactant on another face thereof; a valveinterposed between an oxidant source and said fuel cell to control theflow rate of one of said oxidant and said fuel reactant; a pressuresensor operable to measure pressure of the other of said oxidant andsaid fuel reactant; and an H-infinity controller coupled in a feedbackloop between said valve and said pressure sensor.
 2. The fuel cell ofclaim 1 wherein said pressure sensor measures the pressure of saidoxidant reactant and said valve controls the flow rate of said fuelreactant.
 3. The fuel cell of claim 1 further comprising an integratorwindup compensator in data communication with said H-infinitycontroller.
 4. The fuel cell of claim 3 further comprising a real-timecomputer wherein said H-infinity controller and said integrator windupcompensator are executed in said real-time computer.
 5. The fuel cell ofclaim 1 wherein said H-infinity controller incorporates a control weightmatrix (W_(u)) to provide full performance below a first predeterminedfrequency and reduced performance above a second predeterminedfrequency.
 6. The fuel cell of claim 5 wherein said W_(u) is derivedfrom a control weight matrix transfer function$\left( {W_{u} = \frac{s + 0.1}{{0.1s} + 10}} \right)$ wherein s equalsa complex frequency.
 7. The fuel cell of claim 5 wherein said H-infinitycontroller incorporates said W_(u) according to a response profile of acontrol weight matrix transfer function$\left( {W_{u} = \frac{s + 0.1}{{0.1s} + 10}} \right)$ wherein s equalsa complex frequency.
 8. The fuel cell of claim 1 wherein said H-infinitycontroller incorporates a sensor noise weight matrix (W_(n)) tocompensate for corruption above a first predetermined frequency.
 9. Thefuel cell of claim 8 wherein said W_(n) is derived from a sensor noiseweight matrix transfer function$\left( {W_{n} = \frac{10\left( {s + 10} \right)}{s + 1000}} \right)$wherein s equals a complex frequency.
 10. The fuel cell of claim 8wherein said H-infinity controller incorporates said W_(n) according toa response profile of a sensor noise weight matrix transfer function$\left( {W_{n} = \frac{10\left( {s + 10} \right)}{s + 1000}} \right)$wherein s equals a complex frequency.
 11. The fuel cell of claim 1wherein said H-infinity controller incorporates a performance weightmatrix (W_(e)) to compensate for deviation from a steady-state responsemodel above a first predetermined frequency.
 12. The fuel cell of claim11 wherein said W_(e) is derived from a performance weight matrixtransfer function$\left( {W_{e} = \frac{s + 0.8}{{0.08s} + 0.0008}} \right)$ wherein sequals a complex frequency.
 13. The fuel cell of claim 11 wherein saidH-infinity controller incorporates W_(e) according to a response profileof a performance weight matrix transfer function$\left( {W_{e} = \frac{s + 0.8}{{0.08s} + 0.0008}} \right)$ wherein sequals a complex frequency.
 14. The fuel cell of claim 1 wherein saidH-infinity controller incorporates a control weight matrix (w_(u))according to a response profile of a control weight matrix transferfunction $\left( {W_{u} = \frac{s + 0.1}{{0.1s} + 10}} \right),$ asensor noise weight matrix (W_(n)) according to a response profile of asensor noise weight matrix transfer function$\left( {W_{n} = \frac{10\left( {s + 10} \right)}{s + 1000}} \right),$and a performance weight matrix (W_(e)) according to a response profileof a performance weight matrix transfer function$\left( {W_{e} = \frac{s + 0.8}{{0.08s} + 0.0008}} \right),$ wherein sequals a complex frequency.
 15. A method for operating a fuel cellsystem of the type having a fuel cell with at least one membraneelectrode assembly in reactive interface with an oxidant reactant on oneface thereof and a fuel reactant on another face thereof, said methodcomprising: measuring pressure data of one of said oxidant and said fuelreactant; deriving a setpoint for a flow rate of the other of saidoxidant and said fuel reactor from an H-infinity control model as afunction of said pressure data; and regulating said flow rate based onsaid setpoint.
 16. The method of claim 15 wherein measuring pressuredata measures the pressure of said oxidant reactant, and a setpoint forsaid flow rate of said fuel reactant is derived from said oxidantreactant pressure data.
 17. The method of claim 15 further comprisingthe step of compensating for integrator windup in said H-infinitycontrol model.
 18. The method of claim 17 wherein deriving a setpointand compensating for integrator windup is provided in approximately realtime.
 19. The method of claim 15 wherein deriving a setup includesproviding full performance below a first predetermined frequency andreduced performance above a second predetermined frequency.
 20. Themethod of claim 19 wherein a control weight matrix is derived from acontrol weight matrix transfer function$\left( {W_{u} = \frac{s + 0.1}{{0.1s} + 10}} \right)$ wherein s equalsa complex frequency.
 21. The method of claim 19 wherein said H-infinitycontroller incorporates a control weight matrix according to a responseprofile of a control weight matrix transfer function$\left( {W_{u} = \frac{s + 0.1}{{0.1s} + 10}} \right)$ wherein s equalsa complex frequency.
 22. The method of claim 15 wherein deriving asetpoint includes compensating for noise corruption above apredetermined frequency.
 23. The method of claim 22 wherein a sensornoise weight matrix is derived from a sensor noise weight matrixtransfer function$\left( {W_{n} = \frac{10\left( {s + 10} \right)}{s + 1000}} \right)$wherein s equals a complex frequency.
 24. The method of claim 22 whereinsaid H-infinity controller incorporates a sensor noise weight matrixaccording to a response profile of a sensor noise weight matrix transferfunction$\left( {W_{n} = \frac{10\left( {s + 10} \right)}{s + 1000}} \right)$wherein s equals a complex frequency.
 25. The method of claim 15 whereinderiving a setpoint includes compensating for deviation from asteady-state model above a predetermined frequency.
 26. The method ofclaim 25 wherein a performance weight matrix is derived from aperformance weight matrix transfer function$\left( {W_{e} = \frac{s + 0.8}{{0.08s} + 0.0008}} \right)$ wherein sequals a complex frequency.
 27. The method of claim 25 wherein saidH-infinity controller incorporates a performance weight matrix accordingto a response profile of a performance weight matrix transfer function$\left( {W_{e} = \frac{s + 0.8}{{0.08s} + 0.0008}} \right)$ wherein sequals a complex frequency.
 28. The method of claim 15 wherein saidH-infinity control model incorporates a control weight matrix accordingto a response profile of a control weight matrix transfer function$\left( {W_{u} = \frac{s + 0.1}{{0.1s} + 10}} \right),$ a sensor noiseweight matrix according to a response profile of a sensor noise weightmatrix transfer function$\left( {W_{n} = \frac{10\left( {s + 10} \right)}{s + 1000}} \right),$and a performance weight matrix according to a response profile of aperformance weight matrix transfer function$\left( {W_{e} = \frac{s + 0.8}{{0.08s} + 0.0008}} \right),$ wherein sequals a complex frequency.